Exercises for Cosmology

Prof. Dr. P. Schneider & Dr. Hendrik Hildebrandt

Homework 1 (20th { 24th Oct 2014)

1. Quickies

(a) Explain the cosmological principle!

(b) Use the Friedmann equation to derive the expansion law for an Einstein-de-Sitter universe! (Hint: H(a) = _a

a)

2. Flat rotation curve.

We know that the rotation curve of the Milky Way is at, V (R) const: Assumea spherically-symmetric density distribution (r). Determine the functional form of(r) which yields a at rotation curve?

3. Age of the Universe

Based on the Hubble law v = H0D, we can get a simple rst estimate of theage of the Universe.

(a) Consider a galaxy at distance D whose radial velocity is given by this HubbleLaw, and assume that this velocity was the same throughout cosmic time. Inthis case, at some time in the past the separation was zero, and we can identifythat instant as the Big Bang. Under these assumptions, calculate the currentage of the Universe.

(b) Does it depend on the choice of the galaxy, i.e., the current distanceD? Compareyour result with the age of the oldest stars found in our Galaxy, which is about12 109 yr.

(c) Since no signal can propagate faster than the speed of light c, the age of theUniverse times the speed of light is often called the `size of the visible Universe'.How large is that?

4. Quintessence

From the Friedmann equation and the First Law of Thermodynamics, derive theexpansion equation [i.e. H(a)] for a Universe containing matter, radiation and anadditional substance with an equation-of-state p = wc2, where w is a constant (say,w 2 [1; 0]).

You can nd the exercise sheets, class details, literature, links, etc. on our web page:

http://www.astro.uni-bonn.de/~Cosmo

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